Occupancy modeling focuses on inference about the distribution of organisms over space, using temporal or spatial replication to allow inference about the detection process. Inference based on spatial replication strictly requires that replicates be selected randomly and with replacement, but the importance of these design requirements is not well understood. This paper focuses on an increasingly popular sampling design based on spatial replicates that are not selected randomly and that are expected to exhibit Markovian dependence. We develop two new occupancy models for data collected under this sort of design, one based on an underlying Markov model for spatial dependence and the other based on a trap response model with Markovian detections. We then simulated data under the model for Markovian spatial dependence and fit the data to standard occupancy models and to the two new models. Bias of occupancy estimates was substantial for the standard models, smaller for the new trap response model, and negligible for the new spatial process model. We also fit these models to data from a large-scale tiger occupancy survey recently conducted in Karnataka State, southwestern India. In addition to providing evidence of a positive relationship between tiger occupancy and habitat, model selection statistics and estimates strongly supported the use of the model with Markovian spatial dependence. This new model provides another tool for the decomposition of the detection process, which is sometimes needed for proper estimation and which may also permit interesting biological inferences. In addition to designs employing spatial replication, we note the likely existence of temporal Markovian dependence in many designs using temporal replication. The models developed here will be useful either directly, or with minor extensions, for these designs as well. We believe that these new models represent important additions to the suite of modeling tools now available for occupancy estimation in conservation monitoring. More generally, this work represents a contribution to the topic of cluster sampling for situations in which there is a need for specific modeling (e.g., reflecting dependence) for the distribution of the variable(s) of interest among subunits.